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In a Nut Shell:
Continuity
means that if the point (x,y) changes by a small
amount then
the value of f(x,y ) also
changes by a small amount. It
implies that the surface, f(x,y),
contains no holes or
breaks.
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The formal definition of continuity:
A function of two
variables, x and y, is continuous at (a,b) if
lim f(x,y) = f(a,b)
(x,y) →(a,b)
f(x,y)
is continuous at every point (a,b) in D, its
domain.
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Types of continuous functions:
All polynomials in x and
y are continuous on the plane.
The sums, differences,
products, and quotients of continuous functions are continuous.
A rational function is a
ratio of polynomials. So any
rational function is continuous
on its domain, D.
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